OFFSET
0,6
LINKS
G. C. Greubel, Antidiagonal rows n = 0..50, flattened
FORMULA
T(n, k) = Product_{j=1..n} p(j, k+1), p(n, x) = Sum_{j=0..n} (-1)^j*A053122(n, j)*x^j, and T(n, 0) = n! (square array).
T(n, k) = Product_{j=0..n-1} (-1)^j*ChebyshevU(j, (k-1)/2) with T(n, 0) = n! for n >= 1, and T(0, k) = 1 (square array). - G. C. Greubel, Jun 25 2021
EXAMPLE
Square array begins as:
1, 1, 1, 1, 1, 1, 1 ...;
1, 1, 1, 1, 1, 1, 1 ...;
2, 0, -1, -2, -3, -4, -5 ...;
6, 0, 0, -6, -24, -60, -120 ...;
24, 0, 0, 24, 504, 3360, 13800 ...;
120, 0, 0, 120, 27720, 702240, 7603800 ...;
720, 0, 0, -720, -3991680, -547747200, -20074032000 ...;
Antidiagonal rows begin as:
1;
1, 1;
1, 1, 2;
1, 1, 0, 6;
1, 1, -1, 0, 24;
1, 1, -2, 0, 0, 120;
1, 1, -3, -6, 0, 0, 720;
1, 1, -4, -24, 24, 0, 0, 5040;
1, 1, -5, -60, 504, 120, 0, 0, 40320;
1, 1, -6, -120, 3360, 27720, -720, 0, 0, 362880;
1, 1, -7, -210, 13800, 702240, -3991680, -5040, 0, 0, 3628800;
MATHEMATICA
(* First program *)
b[n_, k_]:= If[k==n, 2, If[k==n-1 || k==n+1, -1, 0]];
M[d_]:= Table[b[n, k], {n, d}, {k, d}];
p[x_, n_]:= If[n==0, 1, CharacteristicPolynomial[M[n], x]];
f = Table[p[x, n], {n, 0, 20}];
T[n_, k_]:= If[k==0, n!, Product[f[[j]], {j, n}]/.x->(k+1)];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jun 25 2021 *)
(* Second program *)
T[n_, k_]:= If[n==0, 1, If[k==0, n!, Product[(-1)^j*Simplify[ChebyshevU[j, x/2-1]], {j, 0, n-1}]/.x->(k+1)]];
Table[T[k, n-k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 25 2021 *)
PROG
(Sage)
@CachedFunction
def T(n, k):
if (n==0): return 1
elif (k==0): return factorial(n)
else: return product( (-1)^j*chebyshev_U(j, (k-1)/2) for j in (0..n-1) )
flatten([[T(k, n-k) for k in (0..n)] for n in (0..10)]) # G. C. Greubel, Jun 25 2021
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Feb 11 2009
EXTENSIONS
Edited by G. C. Greubel, Jun 25 2021
STATUS
approved